Convexity versus average convexity: potential, pmas, the shapley value and simple games

Different propietats of convex and average convex games are developed in terms of unanimity coordinates. A set of necessary and sufficient conditions for the core of balanced games containing the Shapley value are given. Connections between convexity and average convexity, the Shapley value, the potential and population monotonic allocation schemes (PMAS) are shown. Average convex simple games are described. Finally, some differences between convexity and average convexity in relation with other well-known solution concepts of cooperative games are studied.